Base sizes of modules for finite groups of Lie type

A base for a permutation group G ≤ Sym(Ω) is a subset B ⊆ Ω such that the pointwise stabiliser GB of B in G is trivial. The minimal cardinality of a base for G is called the base size and denoted b(G). The study of base sizes for finite permutation groups has a long and rich history dating back at least to work of Bochert [5] in the nineteenth century. Classically, the main motivation for the study of bases was to bound the order of a permutation group in terms of its degree. Just as a linear map φ : V → V on a finite-dimensional vector space V is uniquely determined by its action on a basis for V , so too is an element of a permutation group uniquely determined by its action on a base. In particular, if Ω is finite, then |G| ≤ |Ω| b(G) , so b(G) ≥ log |G|/ log |Ω|. The definition of a base was first formalised by Sims in the 1970s [77], in the context of computational group theory, noting that a small base provides an efficient way of encoding the elements of a given permutation group. There is therefore a focus in the literature on groups with small bases. In general, permuta tion groups do not have small bases relative to their degree. Indeed, Pyber showed in [72] that there exists a universal constant C such that almost all subgroups H ≤ Sn have b(H) > Cn. There are, however, some important families of permutation groups with small bases. Indeed, Halasi, Liebeck and Mar´oti [42] showed that a primitive permutation group G of degree n has b(G) ≤ 2 log |G|/ log n + 24, improving the constants given in [24], which settled a long-standing conjecture of Pyber [72]. Although this upper bound is asymptotically best possible for the class of all primitive groups, there are many examples of primitive groups with smaller bases, and considerable progress towards classifying such groups. In particular, since primitive groups with base size 1 are cyclic, there is significant interest in classifying the primitive groups with base size 2. This thesis is a contribution to this classification for primitive groups of affine type. A primitive group of affine type has the form H = GV , where V is a finite dimensional vector space over a finite field Fr and G ≤ GL(V ) acts irreducibly on V . It has base size 2 if and only if G has base size 1 in its action on V . That is, G has a regular orbit on V . We consider bases of linear groups G ≤ GL(V ) in the case where G has a unique sub normal quasisimple subgroup E(G) of Lie type that acts absolutely irreducibly on V . For cross-characteristic actions, we show that G has a regular orbit on V , with specific excep tions, for which we find the base size. We show that in general, such actions in cross char 3acteristic have b(G) ≤ 4, except for a single exception where b(G) = 5. Moreover, for E(G)/Z(E(G)) ∼= PSLn(r0), with Fr0 a field of the same characteristic as Fr, we show that G has a regular orbit on V , except possibly when (G, V ) is one of a collection of exceptions where b(G) ≤ 5, or V is the natural module for E(G).Open Acces